Moving Interfaces in Crystalline Solids PDF
Preview Moving Interfaces in Crystalline Solids
& SpringerWienNewYork CISM COURSES AND LECTURES Series Editors: The Rectors Manuel Garcia Velarde - Madrid Jean Salengon - Palaiseau Wilhelm Schneider - Wien The Secretary General Bemhard Schrefler - Padua Executive Editor Carlo Tasso - Udine The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences. INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 453 MOVING INTERFACES IN CRYSTALLINE SOLIDS EDITED BY FRANZ DIETER FISCHER MONTANUNIVERSITAT LEOBEN, AUSTRIA AND ERICH SCHMID INSTITUTE FOR MATERIALS SCIENCE, AUSTRIAN ACADEMY OF SCIENCES, LEOBEN, AUSTRIA SpringerWien NewYork The publication of this volume was co-sponsored and co-financed by the UNESCO Venice Office - Regional Bureau for Science in Europe (ROSTE) and its content corresponds to a CISM Advanced Course supported by the same UNESCO Regional Bureau. This volume contains 126 illustrations This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2004 by CISM, Udine Printed in Italy SPIN 11355274 In order to make this volume available as economically and as rapidly as possible the authors' typescripts have been reproduced in their original forms. This method unfortunately has its typographical limitations but it is hoped that they in no way distract the reader. ISBN 3-211-23899-9 SpringerWienNewYork PREFACE Materials with a changing microstructure are a matter of fact in many fields of materials research as well as in the application of materials in diverse fields of technical practice. Such a change may consist in one or more phase transformations in the material, the growth or shrinking of grains, or the appearance of the material in a different geometrical configuration like twinning. Both physical and thermomechanical activities are necessary to drive or stop or, better, to control such a process. The goal of the course "Moving Discontinuities in Crystalline Solids", on which this booklet is based, was to bring together experts from materials physics and materials mechanics to explain the fundamental phenomena of moving interfaces accompanied by the change of the material on both sides of the interface. The application of both classical and modern concepts of materials physics was demonstrated. Furthermore, the findings of continuum mechanics and numerical methods were discussed ranging from Eshelby's derivation of the thermodynamical force on an interface to numerical concepts under development for multiparticle, multicomponent systems. The authors of the various chapters, however, have tried hard not to present simply diverse concepts but to bridge the gaps between the numerous physical and mechanical approaches to this wide field of knowledge. Such a task is not easy and rather new. For many years researchers from mechanics and physics have tended to take different roads which met only by chance. Within the last fifteen years both groups felt that they had to come nearer, and finally, the authors of this booklet have the impression that they have come together. The inherent interdisciplinarity of the subject has been one of the strongest motivations to perform such a course and to write the booklet at hand. Computational methods, like the Monte Carlo Method, Ab-initio Modelling and "Enriched" Finite Elements have contributed a lot to a better understanding of what is behind the change of the microstructure. One may recognize this in the ever increasing number of courses, seminars, conferences and corresponding papers on "Modelling and Simulation". The authors also feel that the common view on the controlling mechanisms for the microstructure will have an increasing impact on industrial application. More effective materials that are better adapted to their respective functions and require shorter development times are more or less a must in the world of today's technology. Some comments are also necessary with respect to the layout of this booklet. Since a group of researchers got together from different fields, very often their symbols and notations differ. A full unification would take too much time although it is a demanding task for the future. Therefore, the authors decided to write each chapter in a self- contained and self-explaining way with a list of notations at the beginning of the chapter. So each chapter has a textbook character starting from the basics and stating carefully the assumptions and limitations of the application of the theoretical framework. The authors try to present examples that are easy to understand. The authors have also performed a mutual reading of their chapters with the goal to bring their own contribution in line with the related chapters. The authors hope that the booklet will provide a sufficient basis for the understanding of this interdisciplinary field of materials mechanics and materials physics and will further mutual understanding. Both researchers and industrial developers should profit from this rather unique presentation of the motion of interfaces in solids. The authors express their thanks to the Director of CISM, Prof M. G. Velarde, for supporting the course with the staff of CISM and for the strong encouragement they received to write this booklet. Finally, the authors are grateful to Prof. C. Tasso for accepting this booklet for publication and his help in the editing. Franz Dieter Fischer CONTENTS Preface Application of Configurational Mechanics to Elastic Solids with Defects and Cracks by R. Kienzler and G. Herrmann 1 Phase Separation in Binary Alloys-Modelling Approaches by P. Fratzl andR. Weinkamer 57 Utilization of the Thermodynamic Extremal Principle for Modelling in Material Science by J. Svoboda 117 Thermodynamics and Kinetics of Phase and Twin Boundaries by F. D. Fischer and N. K. Simha 169 Moving Grain Boundaries During Hot Deformation of Metals: Dynamic Recrystallization by F. Montheillet 203 Application of Configurational Mechanics to Elastic Solids with Defects and Cracks R. Kienzler^ and G. Herrmann^ ^University of Bremen, Bremen, Germany ^Stanford University, Stanford, CA, USA Abstract. Classical mechanics, i. e., in physical space is concerned with forces, stresses and strains and attempts to describe the motion and/or deformation of bodies with mass. In this context, the notions of tractions, trajectories, balance and conservation laws, stability of equilibrium etc. are well established. Mechanics in material space (or configurational mechanics) describes the behaviour of defects (e. g., voids, dislocations, cracks) as they move relatively to the material, in which they find themselves. Concerning this change of configuration, similar notions, as given above, are introduced in material space. After providing the elements of configurational mechanics the method is applied to elastic solids with defects and cracks. In particular, the hole-dislocation interaction problem is discussed and the use of path-independent integrals and local failure criteria in fracture mechanics are demonstrated. List of Notations a abbreviation a crack length Gi material properties (not further specified) A area, mostly used as differential area element dA A cross-sectional area mostly in connection with compressional stiffness EA A action integral b, bi, b Burgers vector, its components and its magnitude b^j components of Eshelby tensor, material momentum B material force in one-dimensional problem B body, domain of integration Cijj^i components of tensor of elasticity C compliance d thickness of bar E Young's modulus E modified Young's modulus EA compressional or axial stiffness E Euler operator R. Kienzler and G. Herrmann fi characteristics (Neutral Action method) /, / vector of body forces and its components fij ? // 5 //^ dimensionless geometry functions correlated with crack-opening modes /, // and/// F,Fj,F (physical) force vector, its components and its magnitude g gravitational acceleration g() function G shear modulus G, Gj material traction vector acting across a surface with normal vector n and its n n components ^ energy release rate ^ Energy release rate of strength-of-material theories h height of bar ^ /i first invariant of stress tensor J material force (J integral) J, JI vector of material forces and its components (components of J integral) Kj, Kjj, Kjjj stress-intensity factors correlated with crack-opening modes, /, // and /// i length of bar / Lagrangian L null Lagrangian L,L^,L vector of material momentums (L integral) its components and 1 = 1^ in plane problems m mass M material virial (M integral) n, rij unit outward normal vector and its components A^ axial force in one-dimensional problem 0(S") terms of (vanishing) order n and higher P,Pi,P i -component vector (current), its components and current in one-dimensional problem pr^"^"* n-th prolongation Q^ characteristics (Noether's formalism) r radial coordinate in polar coordinate system r^ radius of hole s arc length, mostly used as differential arc-length element ds s abbreviation S surface, domain of integration Sf surface of body 5 with prescribed tractions S^ surface of body 5 with prescribed displacements
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